Jun-ichi Segata

On the well-posedness of the generalized Korteweg–de Vries equation in scale-critical Lr-space

The purpose of this paper is to study local and global well-posedness of initial value problem for generalized Korteweg-de Vries (gKdV) equation in ^L^r. We show (large data) local well-posedness, small data global well-posedness, and small data scattering for gKdV equation in the scale critical ^L^r space. A key ingredient is a Stein-Tomas type inequality for the Airy equation, which generalizes usual Strichartz estimates for ^L^r-framework.

Final State Problem for the Cubic Nonlinear Schrödinger Equation with Repulsive Delta Potential

We consider the asymptotic behavior in time of solutions to the cubic nonlinear Schrödinger equation with repulsive delta potential (δ-NLS). We shall prove that for a given small asymptotic profile u ap , there exists a solution u to (δ-NLS) which converges to u ap in L 2(ℝ) as t → ∞. To show this result we exploit the distorted Fourier transform associated to the Schrödinger equation with delta potential.

Refinement of Strichartz Estimates for Airy Equation in Nondiagonal Case and its Application

In this paper, we give an improvement of nondiagonal Strichartz estimates for the Airy equation by using a Morrey-type space. As its applications, we prove the small data scattering and the existence of special nonscattering solutions, which are minimal in a suitable sense, to the mass-subcritical generalized Korteweg--de Vries equation. Especially, the use of a refined nondiagonal estimate removes several technical restrictions on the previous work [S. Masaki and J. Segata, Existence of a Minimal Non-Scattering Solution to the Mass-Subcritical Generalized Korteweg-de Vries Equation, preprint, https://arxiv.org/abs/1602.05331] about the existence of the special non-scattering solution.